Q.48 - The Practice of Statistics for AP

Question

The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. He got 2048 heads. That's a bit more than one-half. Is this evidence that Count Buffon's coin was not balanced? To find out, Luisa decides to perform a significance test. Unfortunately, she made a few errors along the way. Your job is to spot the mistakes and correct them.

$\begin{array}{r}{H}_0:\mu >0.5\\ H_a:\overline{x}=0.5\end{array}$

- Independent 4040(0.5)=2020 and 4040(1-0.5)=2020 are both at least 10 .

- Normal There are at least 40,400 coins in the world.

$t=\frac{0.5-0.507}{\sqrt{\frac{0.5\left(0.5\right)}{4040}}}=-0.89;P_{\text{-value }}=1-0.1867=0.8133$

Reject $H_0$ because the P-value is so large and conclude that the coin is fair.

Step-by-Step Solution

Answer

The coin is fair as there is enough evidence.

Step 1: Introduction

the hypothesis to be a scientific hypothesis, the scientific technique requires that one can test it. Scientists for the most part base scientific speculations on previous observations that can't satisfactorily be explained with the available scientific theories

Step 2: Explanation

coins have been tossed n= $4040$times

Head obtained x = $2048$

Sample proportion $\stackrel{-}{p}=\frac{x}{n}=\frac{2048}{4040}=0.507$

Calculating the null and alternative hypotheses,

$\begin{array}{r}{H}_0:p=0.5\\ H_0:p\ne 0.5\end{array}$

Using, $Z=\frac{\stackrel{-}{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$$=\frac{0.507-0.50}{\sqrt{\frac{0.50\left(0.50\right)}{4040}}}=0.890$

Calculating the p-value,$=2\times P(Z>|z\left|\right)$

$=2\times P(Z>|0.890\left|\right)=0.373$

The p-value is greater than the significance level hence coin is fair as there is enough evidence.